Groups That Act: Orbits, Stabilizers, and Counting

What it means for a group to act

In Volume I a group was a self-contained algebraic object. A group action connects it to the outside world: it is a rule that lets each element of a group G permute the points of a set X. Formally, an action is a map G × X → X, written g·x, satisfying two axioms: the identity acts trivially, e·x = x, and the action respects multiplication, g·(h·x) = (gh)·x. That second axiom is the whole point — it says “do h then g” equals “do gh,” so each g really behaves like a symmetry, and the assignment g ↦ (x ↦ g·x) is a homomorphism from G into the symmetric group Sym(X).

Two structures fall out of any action. The orbit of a point x is everything you can reach from it: G·x = { g·x : g ∈ G }. The orbits partition X — every point lies in exactly one. The stabilizer of x is the set of group elements that fix it: Stab(x) = { g ∈ G : g·x = x }. Crucially, Stab(x) is always a subgroup of G, because the elements fixing x are closed under multiplication and inverses. The whole theory hinges on the tension between how big the orbit is and how big the stabilizer is.

The orbit-stabilizer theorem

Here is the theorem that does the heavy lifting. The orbit-stabilizer theorem says that for a finite group G acting on X and any point x, |G·x| = [G : Stab(x)] = |G| / |Stab(x)|. The size of the orbit equals the index of the stabilizer. The proof is a bijection: send the coset g·Stab(x) to the point g·x. This is well-defined and injective precisely because g·x = h·x iff h⁻¹g fixes x iff g and h lie in the same coset of Stab(x). So orbit elements correspond exactly to cosets — and counting cosets is Lagrange's theorem, which you already know.

One immediate corollary: every orbit size divides |G|. That single fact is the seed of the class equation and the Sylow theorems you will meet later. Let us see it in action with a concrete symmetry group.

Cube rotations acting on faces.

G = rotation group of a cube. We do NOT yet know |G|; let's compute it.
X = the 6 faces. G acts on X (rotations permute faces).

Fix one face, say the TOP face x.
- Orbit G·x: a rotation can carry the top face to any of the 6 faces.
  So |G·x| = 6.
- Stabilizer Stab(x): rotations that keep the top face where it is.
  These are rotations about the vertical axis: 0, 90, 180, 270 degrees.
  So |Stab(x)| = 4.

Orbit-stabilizer:  |G| = |G·x| * |Stab(x)| = 6 * 4 = 24.

The rotation group of the cube has order 24. (In fact G is isomorphic to S_4,
acting on the 4 long diagonals.) We counted the whole group by looking
at ONE face and asking two easy questions.

The standard actions you will reuse forever

Three actions of a group on itself or its own structure are the bread and butter of everything that follows. Learn them as named objects.

1. Left multiplication: g·x = gx, with X = G. This action is free (stabilizers are trivial) and transitive (one orbit). It packages Cayley's theorem — every group embeds in a symmetric group — and underlies the regular representation.
2. Conjugation: g·x = gxg⁻¹, with X = G. Now orbits are conjugacy classes and stabilizers are centralizers. The whole next guide lives inside this one action.
3. Action on cosets: G acts on the left cosets of a subgroup H by g·(aH) = (ga)H. This gives a homomorphism G → S_{[G:H]} whose kernel is the largest normal subgroup inside H — a key trick for proving a group has a nontrivial normal subgroup.